<s>
In	O
set	O
theory	O
,	O
an	O
ordinal	B-Language
number	I-Language
,	O
or	O
ordinal	B-Language
,	O
is	O
a	O
generalization	O
of	O
ordinal	B-Language
numerals	O
(	O
first	O
,	O
second	O
,	O
th	O
,	O
etc	O
.	O
)	O
</s>
<s>
aimed	O
to	O
extend	O
enumeration	B-Language
to	O
infinite	O
sets	O
.	O
</s>
<s>
To	O
extend	O
this	O
process	O
to	O
various	O
infinite	O
sets	O
,	O
ordinal	B-Language
numbers	I-Language
are	O
defined	O
more	O
generally	O
as	O
linearly	O
ordered	O
labels	O
that	O
include	O
the	O
natural	O
numbers	O
and	O
have	O
the	O
property	O
that	O
every	O
set	O
of	O
ordinals	B-Language
has	O
a	O
least	O
element	O
(	O
this	O
is	O
needed	O
for	O
giving	O
a	O
meaning	O
to	O
"	O
the	O
least	O
unused	O
element	O
"	O
)	O
.	O
</s>
<s>
This	O
more	O
general	O
definition	O
allows	O
us	O
to	O
define	O
an	O
ordinal	B-Language
number	I-Language
(	O
omega	B-Language
)	O
that	O
is	O
greater	O
than	O
every	O
natural	O
number	O
,	O
along	O
with	O
ordinal	B-Language
numbers	I-Language
,	O
,	O
etc.	O
,	O
which	O
are	O
even	O
greater	O
than	O
.	O
</s>
<s>
So	O
ordinal	B-Language
numbers	I-Language
exist	O
and	O
are	O
essentially	O
unique	O
.	O
</s>
<s>
Ordinal	B-Language
numbers	I-Language
are	O
distinct	O
from	O
cardinal	O
numbers	O
,	O
which	O
measure	O
the	O
size	O
of	O
sets	O
.	O
</s>
<s>
Although	O
the	O
distinction	O
between	O
ordinals	B-Language
and	O
cardinals	O
is	O
not	O
always	O
apparent	O
on	O
finite	O
sets	O
(	O
one	O
can	O
go	O
from	O
one	O
to	O
the	O
other	O
just	O
by	O
counting	O
labels	O
)	O
,	O
they	O
are	O
very	O
different	O
in	O
the	O
infinite	O
case	O
,	O
where	O
different	O
infinite	O
ordinals	B-Language
can	O
correspond	O
to	O
sets	O
having	O
the	O
same	O
cardinal	O
.	O
</s>
<s>
Like	O
other	O
kinds	O
of	O
numbers	O
,	O
ordinals	B-Language
can	O
be	O
added	O
,	O
multiplied	O
,	O
and	O
exponentiated	O
,	O
although	O
none	O
of	O
these	O
operations	O
are	O
commutative	O
.	O
</s>
<s>
Ordinals	B-Language
were	O
introduced	O
by	O
Georg	O
Cantor	O
in	O
1883	O
in	O
order	O
to	O
accommodate	O
infinite	O
sequences	O
and	O
classify	O
derived	O
sets	O
,	O
which	O
he	O
had	O
previously	O
introduced	O
in	O
1872	O
while	O
studying	O
the	O
uniqueness	O
of	O
trigonometric	O
series	O
.	O
</s>
<s>
When	O
dealing	O
with	O
infinite	O
sets	O
,	O
however	O
,	O
one	O
has	O
to	O
distinguish	O
between	O
the	O
notion	O
of	O
size	O
,	O
which	O
leads	O
to	O
cardinal	O
numbers	O
,	O
and	O
the	O
notion	O
of	O
position	O
,	O
which	O
leads	O
to	O
the	O
ordinal	B-Language
numbers	I-Language
described	O
here	O
.	O
</s>
<s>
This	O
is	O
because	O
while	O
any	O
set	O
has	O
only	O
one	O
size	O
(	O
its	O
cardinality	B-Application
)	O
,	O
there	O
are	O
many	O
nonisomorphic	O
well-orderings	O
of	O
any	O
infinite	O
set	O
,	O
as	O
explained	O
below	O
.	O
</s>
<s>
Whereas	O
the	O
notion	O
of	O
cardinal	O
number	O
is	O
associated	O
with	O
a	O
set	O
with	O
no	O
particular	O
structure	O
on	O
it	O
,	O
the	O
ordinals	B-Language
are	O
intimately	O
linked	O
with	O
the	O
special	O
kind	O
of	O
sets	O
that	O
are	O
called	O
well-ordered	O
.	O
</s>
<s>
Ordinals	B-Language
may	O
be	O
used	O
to	O
label	O
the	O
elements	O
of	O
any	O
given	O
well-ordered	O
set	O
(	O
the	O
smallest	O
element	O
being	O
labelled	O
0	O
,	O
the	O
one	O
after	O
that	O
1	O
,	O
the	O
next	O
one	O
2	O
,	O
"	O
and	O
so	O
on	O
"	O
)	O
,	O
and	O
to	O
measure	O
the	O
"	O
length	O
"	O
of	O
the	O
whole	O
set	O
by	O
the	O
least	O
ordinal	B-Language
that	O
is	O
not	O
a	O
label	O
for	O
an	O
element	O
of	O
the	O
set	O
.	O
</s>
<s>
Any	O
ordinal	B-Language
is	O
defined	O
by	O
the	O
set	O
of	O
ordinals	B-Language
that	O
precede	O
it	O
.	O
</s>
<s>
In	O
fact	O
,	O
the	O
most	O
common	O
definition	O
of	O
ordinals	B-Language
identifies	O
each	O
ordinal	B-Language
as	O
the	O
set	O
of	O
ordinals	B-Language
that	O
precede	O
it	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
ordinal	B-Language
42	O
is	O
generally	O
identified	O
as	O
the	O
set	O
.	O
</s>
<s>
Conversely	O
,	O
any	O
set	O
S	O
of	O
ordinals	B-Language
that	O
is	O
downward	O
closed	O
meaning	O
that	O
for	O
any	O
ordinal	B-Language
α	O
in	O
S	O
and	O
any	O
ordinal	B-Language
β	O
<	O
α	O
,	O
β	O
is	O
also	O
in	O
S	O
is	O
(	O
or	O
can	O
be	O
identified	O
with	O
)	O
an	O
ordinal	B-Language
.	O
</s>
<s>
This	O
definition	O
of	O
ordinals	B-Language
in	O
terms	O
of	O
sets	O
allows	O
for	O
infinite	O
ordinals	B-Language
.	O
</s>
<s>
The	O
smallest	O
infinite	O
ordinal	B-Language
is	O
,	O
which	O
can	O
be	O
identified	O
with	O
the	O
set	O
of	O
natural	O
numbers	O
(	O
so	O
that	O
the	O
ordinal	B-Language
associated	O
with	O
every	O
natural	O
number	O
precedes	O
)	O
.	O
</s>
<s>
Indeed	O
,	O
the	O
set	O
of	O
natural	O
numbers	O
is	O
well-ordered	O
—	O
as	O
is	O
any	O
set	O
of	O
ordinals	B-Language
—	O
and	O
since	O
it	O
is	O
downward	O
closed	O
,	O
it	O
can	O
be	O
identified	O
with	O
the	O
ordinal	B-Language
associated	O
with	O
it	O
.	O
</s>
<s>
Perhaps	O
a	O
clearer	O
intuition	O
of	O
ordinals	B-Language
can	O
be	O
formed	O
by	O
examining	O
a	O
first	O
few	O
of	O
them	O
:	O
as	O
mentioned	O
above	O
,	O
they	O
start	O
with	O
the	O
natural	O
numbers	O
,	O
0	O
,	O
1	O
,	O
2	O
,	O
3	O
,	O
4	O
,	O
5	O
,	O
…	O
After	O
all	O
natural	O
numbers	O
comes	O
the	O
first	B-Language
infinite	I-Language
ordinal	I-Language
,	O
ω	B-Language
,	O
and	O
after	O
that	O
come	O
ω+1	O
,	O
ω+2	O
,	O
ω+3	O
,	O
and	O
so	O
on	O
.	O
</s>
<s>
After	O
all	O
of	O
these	O
come	O
ω·2	O
(	O
which	O
is	O
ω+ω	O
)	O
,	O
ω·2+1	O
,	O
ω·2+2	O
,	O
and	O
so	O
on	O
,	O
then	O
ω·3	O
,	O
and	O
then	O
later	O
on	O
ω·4	O
.	O
</s>
<s>
Now	O
the	O
set	O
of	O
ordinals	B-Language
formed	O
in	O
this	O
way	O
(	O
the	O
ω·m+n	O
,	O
where	O
m	O
and	O
n	O
are	O
natural	O
numbers	O
)	O
must	O
itself	O
have	O
an	O
ordinal	B-Language
associated	O
with	O
it	O
:	O
and	O
that	O
is	O
ω2	O
.	O
</s>
<s>
Further	O
on	O
,	O
there	O
will	O
be	O
ω3	O
,	O
then	O
ω4	O
,	O
and	O
so	O
on	O
,	O
and	O
ωω	O
,	O
then	O
ωωω	O
,	O
then	O
later	O
ωωωω	O
,	O
and	O
even	O
later	O
ε0	O
(	O
epsilon	O
nought	O
)	O
(	O
to	O
give	O
a	O
few	O
examples	O
of	O
relatively	O
small	O
—	O
countable	O
—	O
ordinals	B-Language
)	O
.	O
</s>
<s>
This	O
can	O
be	O
continued	O
indefinitely	O
(	O
as	O
every	O
time	O
one	O
says	O
"	O
and	O
so	O
on	O
"	O
when	O
enumerating	O
ordinals	B-Language
,	O
it	O
defines	O
a	O
larger	O
ordinal	B-Language
)	O
.	O
</s>
<s>
The	O
smallest	O
uncountable	O
ordinal	B-Language
is	O
the	O
set	O
of	O
all	O
countable	B-Language
ordinals	I-Language
,	O
expressed	O
as	O
ω1	O
or	O
.	O
</s>
<s>
In	O
practice	O
,	O
the	O
importance	O
of	O
well-ordering	O
is	O
justified	O
by	O
the	O
possibility	O
of	O
applying	O
transfinite	B-Algorithm
induction	I-Algorithm
,	O
which	O
says	O
,	O
essentially	O
,	O
that	O
any	O
property	O
that	O
passes	O
on	O
from	O
the	O
predecessors	O
of	O
an	O
element	O
to	O
that	O
element	O
itself	O
must	O
be	O
true	O
of	O
all	O
elements	O
(	O
of	O
the	O
given	O
well-ordered	O
set	O
)	O
.	O
</s>
<s>
Such	O
a	O
one-to-one	B-Algorithm
correspondence	I-Algorithm
is	O
called	O
an	O
order	O
isomorphism	O
,	O
and	O
the	O
two	O
well-ordered	O
sets	O
are	O
said	O
to	O
be	O
order-isomorphic	O
or	O
similar	O
(	O
with	O
the	O
understanding	O
that	O
this	O
is	O
an	O
equivalence	O
relation	O
)	O
.	O
</s>
<s>
Formally	O
,	O
if	O
a	O
partial	O
order	O
≤	O
is	O
defined	O
on	O
the	O
set	O
S	O
,	O
and	O
a	O
partial	O
order	O
≤	O
 '	O
is	O
defined	O
on	O
the	O
set	O
S	O
 '	O
,	O
then	O
the	O
posets	O
(	O
S	O
,	O
≤	O
)	O
and	O
(	O
S	O
 '	O
,	O
≤	O
 '	O
)	O
are	O
order	O
isomorphic	O
if	O
there	O
is	O
a	O
bijection	B-Algorithm
f	O
that	O
preserves	O
the	O
ordering	O
.	O
</s>
<s>
This	O
is	O
exactly	O
what	O
the	O
ordinals	B-Language
provide	O
,	O
and	O
it	O
also	O
provides	O
a	O
canonical	O
labeling	O
of	O
the	O
elements	O
of	O
any	O
well-ordered	O
set	O
.	O
</s>
<s>
Every	O
well-ordered	O
set	O
(	O
S	O
,	O
<	O
)	O
is	O
order-isomorphic	O
to	O
the	O
set	O
of	O
ordinals	B-Language
less	O
than	O
one	O
specific	O
ordinal	B-Language
number	I-Language
under	O
their	O
natural	O
ordering	O
.	O
</s>
<s>
Essentially	O
,	O
an	O
ordinal	B-Language
is	O
intended	O
to	O
be	O
defined	O
as	O
an	O
isomorphism	O
class	O
of	O
well-ordered	O
sets	O
:	O
that	O
is	O
,	O
as	O
an	O
equivalence	O
class	O
for	O
the	O
equivalence	O
relation	O
of	O
"	O
being	O
order-isomorphic	O
"	O
.	O
</s>
<s>
The	O
ordinal	B-Language
can	O
be	O
said	O
to	O
be	O
the	O
order	O
type	O
of	O
any	O
set	O
in	O
the	O
class	O
.	O
</s>
<s>
The	O
original	O
definition	O
of	O
ordinal	B-Language
numbers	I-Language
,	O
found	O
for	O
example	O
in	O
the	O
Principia	O
Mathematica	O
,	O
defines	O
the	O
order	O
type	O
of	O
a	O
well-ordering	O
as	O
the	O
set	O
of	O
all	O
well-orderings	O
similar	O
(	O
order-isomorphic	O
)	O
to	O
that	O
well-ordering	O
:	O
in	O
other	O
words	O
,	O
an	O
ordinal	B-Language
number	I-Language
is	O
genuinely	O
an	O
equivalence	O
class	O
of	O
well-ordered	O
sets	O
.	O
</s>
<s>
However	O
,	O
this	O
definition	O
still	O
can	O
be	O
used	O
in	O
type	O
theory	O
and	O
in	O
Quine	O
's	O
axiomatic	O
set	O
theory	O
New	O
Foundations	O
and	O
related	O
systems	O
(	O
where	O
it	O
affords	O
a	O
rather	O
surprising	O
alternative	O
solution	O
to	O
the	O
Burali-Forti	O
paradox	O
of	O
the	O
largest	O
ordinal	B-Language
)	O
.	O
</s>
<s>
Rather	O
than	O
defining	O
an	O
ordinal	B-Language
as	O
an	O
equivalence	O
class	O
of	O
well-ordered	O
sets	O
,	O
it	O
will	O
be	O
defined	O
as	O
a	O
particular	O
well-ordered	O
set	O
that	O
(	O
canonically	O
)	O
represents	O
the	O
class	O
.	O
</s>
<s>
Thus	O
,	O
an	O
ordinal	B-Language
number	I-Language
will	O
be	O
a	O
well-ordered	O
set	O
;	O
and	O
every	O
well-ordered	O
set	O
will	O
be	O
order-isomorphic	O
to	O
exactly	O
one	O
ordinal	B-Language
number	I-Language
.	O
</s>
<s>
This	O
motivates	O
the	O
standard	O
definition	O
,	O
suggested	O
by	O
John	O
von	O
Neumann	O
at	O
the	O
age	O
of	O
19	O
,	O
now	O
called	O
definition	O
of	O
von	O
Neumann	O
ordinals	B-Language
:	O
"	O
each	O
ordinal	B-Language
is	O
the	O
well-ordered	O
set	O
of	O
all	O
smaller	O
ordinals.	O
"	O
</s>
<s>
A	O
set	O
S	O
is	O
an	O
ordinal	B-Language
if	O
and	O
only	O
if	O
S	O
is	O
strictly	O
well-ordered	O
with	O
respect	O
to	O
set	O
membership	O
and	O
every	O
element	O
of	O
S	O
is	O
also	O
a	O
subset	O
of	O
S	O
.	O
</s>
<s>
The	O
natural	O
numbers	O
are	O
thus	O
ordinals	B-Language
by	O
this	O
definition	O
.	O
</s>
<s>
It	O
can	O
be	O
shown	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
that	O
every	O
well-ordered	O
set	O
is	O
order-isomorphic	O
to	O
exactly	O
one	O
of	O
these	O
ordinals	B-Language
,	O
that	O
is	O
,	O
there	O
is	O
an	O
order	O
preserving	O
bijective	B-Algorithm
function	I-Algorithm
between	O
them	O
.	O
</s>
<s>
Furthermore	O
,	O
the	O
elements	O
of	O
every	O
ordinal	B-Language
are	O
ordinals	B-Language
themselves	O
.	O
</s>
<s>
Given	O
two	O
ordinals	B-Language
S	O
and	O
T	O
,	O
S	O
is	O
an	O
element	O
of	O
T	O
if	O
and	O
only	O
if	O
S	O
is	O
a	O
proper	O
subset	O
of	O
T	O
.	O
Moreover	O
,	O
either	O
S	O
is	O
an	O
element	O
of	O
T	O
,	O
or	O
T	O
is	O
an	O
element	O
of	O
S	O
,	O
or	O
they	O
are	O
equal	O
.	O
</s>
<s>
So	O
every	O
set	O
of	O
ordinals	B-Language
is	O
totally	O
ordered	O
.	O
</s>
<s>
Further	O
,	O
every	O
set	O
of	O
ordinals	B-Language
is	O
well-ordered	O
.	O
</s>
<s>
Consequently	O
,	O
every	O
ordinal	B-Language
S	O
is	O
a	O
set	O
having	O
as	O
elements	O
precisely	O
the	O
ordinals	B-Language
smaller	O
than	O
S	O
.	O
For	O
example	O
,	O
every	O
set	O
of	O
ordinals	B-Language
has	O
a	O
supremum	O
,	O
the	O
ordinal	B-Language
obtained	O
by	O
taking	O
the	O
union	O
of	O
all	O
the	O
ordinals	B-Language
in	O
the	O
set	O
.	O
</s>
<s>
The	O
class	O
of	O
all	O
ordinals	B-Language
is	O
not	O
a	O
set	O
.	O
</s>
<s>
If	O
it	O
were	O
a	O
set	O
,	O
one	O
could	O
show	O
that	O
it	O
was	O
an	O
ordinal	B-Language
and	O
thus	O
a	O
member	O
of	O
itself	O
,	O
which	O
would	O
contradict	O
its	O
strict	O
ordering	O
by	O
membership	O
.	O
</s>
<s>
The	O
class	O
of	O
all	O
ordinals	B-Language
is	O
variously	O
called	O
"	O
Ord	O
"	O
,	O
"	O
ON	O
"	O
,	O
or	O
"	O
∞	O
"	O
.	O
</s>
<s>
An	O
ordinal	B-Language
is	O
finite	O
if	O
and	O
only	O
if	O
the	O
opposite	O
order	O
is	O
also	O
well-ordered	O
,	O
which	O
is	O
the	O
case	O
if	O
and	O
only	O
if	O
each	O
of	O
its	O
non-empty	O
subsets	O
has	O
a	O
maximum	O
.	O
</s>
<s>
There	O
are	O
other	O
modern	O
formulations	O
of	O
the	O
definition	O
of	O
ordinal	B-Language
.	O
</s>
<s>
For	O
example	O
,	O
assuming	O
the	O
axiom	B-General_Concept
of	I-General_Concept
regularity	I-General_Concept
,	O
the	O
following	O
are	O
equivalent	O
for	O
a	O
set	O
x	O
:	O
</s>
<s>
x	O
is	O
a	O
(	O
von	O
Neumann	O
)	O
ordinal	B-Language
,	O
</s>
<s>
In	O
set	O
theories	O
with	O
urelements	O
,	O
one	O
has	O
to	O
further	O
make	O
sure	O
that	O
the	O
definition	O
excludes	O
urelements	O
from	O
appearing	O
in	O
ordinals	B-Language
.	O
</s>
<s>
If	O
α	O
is	O
any	O
ordinal	B-Language
and	O
X	O
is	O
a	O
set	O
,	O
an	O
α-indexed	O
sequence	O
of	O
elements	O
of	O
X	O
is	O
a	O
function	O
from	O
α	O
to	O
X	O
.	O
</s>
<s>
This	O
concept	O
,	O
a	O
transfinite	O
sequence	O
(	O
if	O
α	O
is	O
infinite	O
)	O
or	O
ordinal-indexed	O
sequence	O
,	O
is	O
a	O
generalization	O
of	O
the	O
concept	O
of	O
a	O
sequence	O
.	O
</s>
<s>
An	O
ordinary	O
sequence	O
corresponds	O
to	O
the	O
case	O
α	O
=	O
ω	B-Language
,	O
while	O
a	O
finite	O
α	O
corresponds	O
to	O
a	O
tuple	O
,	O
a.k.a.	O
</s>
<s>
Transfinite	B-Algorithm
induction	I-Algorithm
holds	O
in	O
any	O
well-ordered	O
set	O
,	O
but	O
it	O
is	O
so	O
important	O
in	O
relation	O
to	O
ordinals	B-Language
that	O
it	O
is	O
worth	O
restating	O
here	O
.	O
</s>
<s>
Any	O
property	O
that	O
passes	O
from	O
the	O
set	O
of	O
ordinals	B-Language
smaller	O
than	O
a	O
given	O
ordinal	B-Language
α	O
to	O
α	O
itself	O
,	O
is	O
true	O
of	O
all	O
ordinals	B-Language
.	O
</s>
<s>
Or	O
,	O
more	O
practically	O
:	O
in	O
order	O
to	O
prove	O
a	O
property	O
P	O
for	O
all	O
ordinals	B-Language
α	O
,	O
one	O
can	O
assume	O
that	O
it	O
is	O
already	O
known	O
for	O
all	O
smaller	O
.	O
</s>
<s>
Transfinite	B-Algorithm
induction	I-Algorithm
can	O
be	O
used	O
not	O
only	O
to	O
prove	O
things	O
,	O
but	O
also	O
to	O
define	O
them	O
.	O
</s>
<s>
Such	O
a	O
definition	O
is	O
normally	O
said	O
to	O
be	O
by	O
transfinite	O
recursion	O
–	O
the	O
proof	O
that	O
the	O
result	O
is	O
well-defined	O
uses	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Let	O
F	O
denote	O
a	O
(	O
class	O
)	O
function	O
F	O
to	O
be	O
defined	O
on	O
the	O
ordinals	B-Language
.	O
</s>
<s>
The	O
idea	O
now	O
is	O
that	O
,	O
in	O
defining	O
F(α )	O
for	O
an	O
unspecified	O
ordinal	B-Language
α	O
,	O
one	O
may	O
assume	O
that	O
F(β )	O
is	O
already	O
defined	O
for	O
all	O
and	O
thus	O
give	O
a	O
formula	O
for	O
F(α )	O
in	O
terms	O
of	O
these	O
F(β )	O
.	O
</s>
<s>
It	O
then	O
follows	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
that	O
there	O
is	O
one	O
and	O
only	O
one	O
function	O
satisfying	O
the	O
recursion	O
formula	O
up	O
to	O
and	O
including	O
α	O
.	O
</s>
<s>
Here	O
is	O
an	O
example	O
of	O
definition	O
by	O
transfinite	O
recursion	O
on	O
the	O
ordinals	B-Language
(	O
more	O
will	O
be	O
given	O
later	O
)	O
:	O
define	O
function	O
F	O
by	O
letting	O
F(α )	O
be	O
the	O
smallest	O
ordinal	B-Language
not	O
in	O
the	O
set	O
,	O
that	O
is	O
,	O
the	O
set	O
consisting	O
of	O
all	O
F(β )	O
for	O
.	O
</s>
<s>
In	O
fact	O
,	O
F(0 )	O
makes	O
sense	O
since	O
there	O
is	O
no	O
ordinal	B-Language
,	O
and	O
the	O
set	O
is	O
empty	O
.	O
</s>
<s>
So	O
F(0 )	O
is	O
equal	O
to	O
0	O
(	O
the	O
smallest	O
ordinal	B-Language
of	O
all	O
)	O
.	O
</s>
<s>
Now	O
that	O
F(0 )	O
is	O
known	O
,	O
the	O
definition	O
applied	O
to	O
F(1 )	O
makes	O
sense	O
(	O
it	O
is	O
the	O
smallest	O
ordinal	B-Language
not	O
in	O
the	O
singleton	O
set	O
)	O
,	O
and	O
so	O
on	O
(	O
the	O
and	O
so	O
on	O
is	O
exactly	O
transfinite	B-Algorithm
induction	I-Algorithm
)	O
.	O
</s>
<s>
It	O
turns	O
out	O
that	O
this	O
example	O
is	O
not	O
very	O
exciting	O
,	O
since	O
provably	O
for	O
all	O
ordinals	B-Language
α	O
,	O
which	O
can	O
be	O
shown	O
,	O
precisely	O
,	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
.	O
</s>
<s>
Any	O
nonzero	O
ordinal	B-Language
has	O
the	O
minimum	O
element	O
,	O
zero	O
.	O
</s>
<s>
For	O
example	O
,	O
42	O
has	O
maximum	O
41	O
and	O
ω+6	O
has	O
maximum	O
ω+5	O
.	O
</s>
<s>
On	O
the	O
other	O
hand	O
,	O
ω	B-Language
does	O
not	O
have	O
a	O
maximum	O
since	O
there	O
is	O
no	O
largest	O
natural	O
number	O
.	O
</s>
<s>
If	O
an	O
ordinal	B-Language
has	O
a	O
maximum	O
α	O
,	O
then	O
it	O
is	O
the	O
next	O
ordinal	B-Language
after	O
α	O
,	O
and	O
it	O
is	O
called	O
a	O
successor	O
ordinal	B-Language
,	O
namely	O
the	O
successor	O
of	O
α	O
,	O
written	O
α+1	O
.	O
</s>
<s>
In	O
the	O
von	O
Neumann	O
definition	O
of	O
ordinals	B-Language
,	O
the	O
successor	O
of	O
α	O
is	O
since	O
its	O
elements	O
are	O
those	O
of	O
α	O
and	O
α	O
itself	O
.	O
</s>
<s>
A	O
nonzero	O
ordinal	B-Language
that	O
is	O
not	O
a	O
successor	O
is	O
called	O
a	O
limit	O
ordinal	B-Language
.	O
</s>
<s>
One	O
justification	O
for	O
this	O
term	O
is	O
that	O
a	O
limit	O
ordinal	B-Language
is	O
the	O
limit	O
in	O
a	O
topological	O
sense	O
of	O
all	O
smaller	O
ordinals	B-Language
(	O
under	O
the	O
order	B-Algorithm
topology	I-Algorithm
)	O
.	O
</s>
<s>
When	O
is	O
an	O
ordinal-indexed	O
sequence	O
,	O
indexed	O
by	O
a	O
limit	O
and	O
the	O
sequence	O
is	O
increasing	O
,	O
i.e.	O
</s>
<s>
whenever	O
its	O
limit	O
is	O
defined	O
as	O
the	O
least	O
upper	O
bound	O
of	O
the	O
set	O
that	O
is	O
,	O
the	O
smallest	O
ordinal	B-Language
(	O
it	O
always	O
exists	O
)	O
greater	O
than	O
any	O
term	O
of	O
the	O
sequence	O
.	O
</s>
<s>
In	O
this	O
sense	O
,	O
a	O
limit	O
ordinal	B-Language
is	O
the	O
limit	O
of	O
all	O
smaller	O
ordinals	B-Language
(	O
indexed	O
by	O
itself	O
)	O
.	O
</s>
<s>
Put	O
more	O
directly	O
,	O
it	O
is	O
the	O
supremum	O
of	O
the	O
set	O
of	O
smaller	O
ordinals	B-Language
.	O
</s>
<s>
Another	O
way	O
of	O
defining	O
a	O
limit	O
ordinal	B-Language
is	O
to	O
say	O
that	O
α	O
is	O
a	O
limit	O
ordinal	B-Language
if	O
and	O
only	O
if	O
:	O
</s>
<s>
There	O
is	O
an	O
ordinal	B-Language
less	O
than	O
α	O
and	O
whenever	O
ζ	O
is	O
an	O
ordinal	B-Language
less	O
than	O
α	O
,	O
then	O
there	O
exists	O
an	O
ordinal	B-Language
ξ	O
such	O
that	O
ζξα	O
.	O
</s>
<s>
ω	B-Language
is	O
a	O
limit	O
ordinal	B-Language
because	O
for	O
any	O
smaller	O
ordinal	B-Language
(	O
in	O
this	O
example	O
,	O
a	O
natural	O
number	O
)	O
there	O
is	O
another	O
ordinal	B-Language
(	O
natural	O
number	O
)	O
larger	O
than	O
it	O
,	O
but	O
still	O
less	O
than	O
ω	B-Language
.	O
</s>
<s>
Thus	O
,	O
every	O
ordinal	B-Language
is	O
either	O
zero	O
,	O
or	O
a	O
successor	O
(	O
of	O
a	O
well-defined	O
predecessor	O
)	O
,	O
or	O
a	O
limit	O
.	O
</s>
<s>
Very	O
often	O
,	O
when	O
defining	O
a	O
function	O
F	O
by	O
transfinite	O
recursion	O
on	O
all	O
ordinals	B-Language
,	O
one	O
defines	O
F(0 )	O
,	O
and	O
F( α+1	O
)	O
assuming	O
F(α )	O
is	O
defined	O
,	O
and	O
then	O
,	O
for	O
limit	O
ordinals	B-Language
δ	O
one	O
defines	O
F(δ )	O
as	O
the	O
limit	O
of	O
the	O
F(β )	O
for	O
all	O
β	O
<	O
δ	O
(	O
either	O
in	O
the	O
sense	O
of	O
ordinal	B-Language
limits	O
,	O
as	O
previously	O
explained	O
,	O
or	O
for	O
some	O
other	O
notion	O
of	O
limit	O
if	O
F	O
does	O
not	O
take	O
ordinal	B-Language
values	O
)	O
.	O
</s>
<s>
Thus	O
,	O
the	O
interesting	O
step	O
in	O
the	O
definition	O
is	O
the	O
successor	O
step	O
,	O
not	O
the	O
limit	O
ordinals	B-Language
.	O
</s>
<s>
Such	O
functions	O
(	O
especially	O
for	O
F	O
nondecreasing	O
and	O
taking	O
ordinal	B-Language
values	O
)	O
are	O
called	O
continuous	O
.	O
</s>
<s>
Ordinal	B-Language
addition	O
,	O
multiplication	O
and	O
exponentiation	O
are	O
continuous	O
as	O
functions	O
of	O
their	O
second	O
argument	O
(	O
but	O
can	O
be	O
defined	O
non-recursively	O
)	O
.	O
</s>
<s>
Any	O
well-ordered	O
set	O
is	O
similar	O
(	O
order-isomorphic	O
)	O
to	O
a	O
unique	O
ordinal	B-Language
number	I-Language
;	O
in	O
other	O
words	O
,	O
its	O
elements	O
can	O
be	O
indexed	O
in	O
increasing	O
fashion	O
by	O
the	O
ordinals	B-Language
less	O
than	O
.	O
</s>
<s>
This	O
applies	O
,	O
in	O
particular	O
,	O
to	O
any	O
set	O
of	O
ordinals	B-Language
:	O
any	O
set	O
of	O
ordinals	B-Language
is	O
naturally	O
indexed	O
by	O
the	O
ordinals	B-Language
less	O
than	O
some	O
.	O
</s>
<s>
The	O
same	O
holds	O
,	O
with	O
a	O
slight	O
modification	O
,	O
for	O
classes	O
of	O
ordinals	B-Language
(	O
a	O
collection	O
of	O
ordinals	B-Language
,	O
possibly	O
too	O
large	O
to	O
form	O
a	O
set	O
,	O
defined	O
by	O
some	O
property	O
)	O
:	O
any	O
class	O
of	O
ordinals	B-Language
can	O
be	O
indexed	O
by	O
ordinals	B-Language
(	O
and	O
,	O
when	O
the	O
class	O
is	O
unbounded	O
in	O
the	O
class	O
of	O
all	O
ordinals	B-Language
,	O
this	O
puts	O
it	O
in	O
class-bijection	O
with	O
the	O
class	O
of	O
all	O
ordinals	B-Language
)	O
.	O
</s>
<s>
Formally	O
,	O
the	O
definition	O
is	O
by	O
transfinite	B-Algorithm
induction	I-Algorithm
:	O
the	O
-th	O
element	O
of	O
the	O
class	O
is	O
defined	O
(	O
provided	O
it	O
has	O
already	O
been	O
defined	O
for	O
all	O
)	O
,	O
as	O
the	O
smallest	O
element	O
greater	O
than	O
the	O
-th	O
element	O
for	O
all	O
.	O
</s>
<s>
This	O
could	O
be	O
applied	O
,	O
for	O
example	O
,	O
to	O
the	O
class	O
of	O
limit	O
ordinals	B-Language
:	O
the	O
-th	O
ordinal	B-Language
,	O
which	O
is	O
either	O
a	O
limit	O
or	O
zero	O
is	O
(	O
see	O
ordinal	B-Language
arithmetic	O
for	O
the	O
definition	O
of	O
multiplication	O
of	O
ordinals	B-Language
)	O
.	O
</s>
<s>
Similarly	O
,	O
one	O
can	O
consider	O
additively	O
indecomposable	O
ordinals	B-Language
(	O
meaning	O
a	O
nonzero	O
ordinal	B-Language
that	O
is	O
not	O
the	O
sum	O
of	O
two	O
strictly	O
smaller	O
ordinals	B-Language
)	O
:	O
the	O
-th	O
additively	O
indecomposable	O
ordinal	B-Language
is	O
indexed	O
as	O
.	O
</s>
<s>
The	O
technique	O
of	O
indexing	O
classes	O
of	O
ordinals	B-Language
is	O
often	O
useful	O
in	O
the	O
context	O
of	O
fixed	O
points	O
:	O
for	O
example	O
,	O
the	O
-th	O
ordinal	B-Language
such	O
that	O
is	O
written	O
.	O
</s>
<s>
A	O
class	O
of	O
ordinals	B-Language
is	O
said	O
to	O
be	O
unbounded	O
,	O
or	O
cofinal	O
,	O
when	O
given	O
any	O
ordinal	B-Language
,	O
there	O
is	O
a	O
in	O
such	O
that	O
(	O
then	O
the	O
class	O
must	O
be	O
a	O
proper	O
class	O
,	O
i.e.	O
,	O
it	O
cannot	O
be	O
a	O
set	O
)	O
.	O
</s>
<s>
It	O
is	O
said	O
to	O
be	O
closed	O
when	O
the	O
limit	O
of	O
a	O
sequence	O
of	O
ordinals	B-Language
in	O
the	O
class	O
is	O
again	O
in	O
the	O
class	O
:	O
or	O
,	O
equivalently	O
,	O
when	O
the	O
indexing	O
(	O
class	O
-	O
)	O
function	O
is	O
continuous	O
in	O
the	O
sense	O
that	O
,	O
for	O
a	O
limit	O
ordinal	B-Language
,	O
(	O
the	O
-th	O
ordinal	B-Language
in	O
the	O
class	O
)	O
is	O
the	O
limit	O
of	O
all	O
for	O
;	O
this	O
is	O
also	O
the	O
same	O
as	O
being	O
closed	O
,	O
in	O
the	O
topological	O
sense	O
,	O
for	O
the	O
order	B-Algorithm
topology	I-Algorithm
(	O
to	O
avoid	O
talking	O
of	O
topology	O
on	O
proper	O
classes	O
,	O
one	O
can	O
demand	O
that	O
the	O
intersection	O
of	O
the	O
class	O
with	O
any	O
given	O
ordinal	B-Language
is	O
closed	O
for	O
the	O
order	B-Algorithm
topology	I-Algorithm
on	O
that	O
ordinal	B-Language
,	O
this	O
is	O
again	O
equivalent	O
)	O
.	O
</s>
<s>
Of	O
particular	O
importance	O
are	O
those	O
classes	O
of	O
ordinals	B-Language
that	O
are	O
closed	O
and	O
unbounded	O
,	O
sometimes	O
called	O
clubs	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
class	O
of	O
all	O
limit	O
ordinals	B-Language
is	O
closed	O
and	O
unbounded	O
:	O
this	O
translates	O
the	O
fact	O
that	O
there	O
is	O
always	O
a	O
limit	O
ordinal	B-Language
greater	O
than	O
a	O
given	O
ordinal	B-Language
,	O
and	O
that	O
a	O
limit	O
of	O
limit	O
ordinals	B-Language
is	O
a	O
limit	O
ordinal	B-Language
(	O
a	O
fortunate	O
fact	O
if	O
the	O
terminology	O
is	O
to	O
make	O
any	O
sense	O
at	O
all	O
!	O
)	O
.	O
</s>
<s>
The	O
class	O
of	O
additively	O
indecomposable	O
ordinals	B-Language
,	O
or	O
the	O
class	O
of	O
ordinals	B-Language
,	O
or	O
the	O
class	O
of	O
cardinals	O
,	O
are	O
all	O
closed	O
unbounded	O
;	O
the	O
set	O
of	O
regular	O
cardinals	O
,	O
however	O
,	O
is	O
unbounded	O
but	O
not	O
closed	O
,	O
and	O
any	O
finite	O
set	O
of	O
ordinals	B-Language
is	O
closed	O
but	O
not	O
unbounded	O
.	O
</s>
<s>
All	O
superclasses	O
of	O
closed	O
unbounded	O
classes	O
are	O
stationary	O
,	O
and	O
stationary	O
classes	O
are	O
unbounded	O
,	O
but	O
there	O
are	O
stationary	O
classes	O
that	O
are	O
not	O
closed	O
and	O
stationary	O
classes	O
that	O
have	O
no	O
closed	O
unbounded	O
subclass	O
(	O
such	O
as	O
the	O
class	O
of	O
all	O
limit	O
ordinals	B-Language
with	O
countable	O
cofinality	O
)	O
.	O
</s>
<s>
the	O
class	O
of	O
ordinals	B-Language
with	O
cofinality	O
ω	B-Language
with	O
the	O
class	O
of	O
ordinals	B-Language
with	O
uncountable	O
cofinality	O
.	O
</s>
<s>
Rather	O
than	O
formulating	O
these	O
definitions	O
for	O
(	O
proper	O
)	O
classes	O
of	O
ordinals	B-Language
,	O
one	O
can	O
formulate	O
them	O
for	O
sets	O
of	O
ordinals	B-Language
below	O
a	O
given	O
ordinal	B-Language
:	O
A	O
subset	O
of	O
a	O
limit	O
ordinal	B-Language
is	O
said	O
to	O
be	O
unbounded	O
(	O
or	O
cofinal	O
)	O
under	O
provided	O
any	O
ordinal	B-Language
less	O
than	O
is	O
less	O
than	O
some	O
ordinal	B-Language
in	O
the	O
set	O
.	O
</s>
<s>
More	O
generally	O
,	O
one	O
can	O
call	O
a	O
subset	O
of	O
any	O
ordinal	B-Language
cofinal	O
in	O
provided	O
every	O
ordinal	B-Language
less	O
than	O
is	O
less	O
than	O
or	O
equal	O
to	O
some	O
ordinal	B-Language
in	O
the	O
set	O
.	O
</s>
<s>
The	O
subset	O
is	O
said	O
to	O
be	O
closed	O
under	O
provided	O
it	O
is	O
closed	O
for	O
the	O
order	B-Algorithm
topology	I-Algorithm
in	O
,	O
i.e.	O
</s>
<s>
a	O
limit	O
of	O
ordinals	B-Language
in	O
the	O
set	O
is	O
either	O
in	O
the	O
set	O
or	O
equal	O
to	O
itself	O
.	O
</s>
<s>
There	O
are	O
three	O
usual	O
operations	O
on	O
ordinals	B-Language
:	O
addition	O
,	O
multiplication	O
,	O
and	O
(	O
ordinal	B-Language
)	O
exponentiation	O
.	O
</s>
<s>
The	O
Cantor	O
normal	O
form	O
provides	O
a	O
standardized	O
way	O
of	O
writing	O
ordinals	B-Language
.	O
</s>
<s>
It	O
uniquely	O
represents	O
each	O
ordinal	B-Language
as	O
a	O
finite	O
sum	O
of	O
ordinal	B-Language
powers	O
of	O
ω	B-Language
.	O
</s>
<s>
However	O
,	O
this	O
cannot	O
form	O
the	O
basis	O
of	O
a	O
universal	O
ordinal	B-Language
notation	O
due	O
to	O
such	O
self-referential	O
representations	O
as	O
ε0	O
=	O
ωε0	O
.	O
</s>
<s>
Interpreted	O
as	O
nimbers	O
(	O
a	O
game-theoretic	O
variant	O
of	O
numbers	O
)	O
,	O
ordinals	B-Language
are	O
also	O
subject	O
to	O
nimber	O
arithmetic	O
operations	O
.	O
</s>
<s>
Each	O
ordinal	B-Language
associates	O
with	O
one	O
cardinal	O
,	O
its	O
cardinality	B-Application
.	O
</s>
<s>
If	O
there	O
is	O
a	O
bijection	B-Algorithm
between	O
two	O
ordinals	B-Language
(	O
e.g.	O
</s>
<s>
Any	O
well-ordered	O
set	O
having	O
an	O
ordinal	B-Language
as	O
its	O
order-type	O
has	O
the	O
same	O
cardinality	B-Application
as	O
that	O
ordinal	B-Language
.	O
</s>
<s>
The	O
least	O
ordinal	B-Language
associated	O
with	O
a	O
given	O
cardinal	O
is	O
called	O
the	O
initial	O
ordinal	B-Language
of	O
that	O
cardinal	O
.	O
</s>
<s>
Every	O
finite	O
ordinal	B-Language
(	O
natural	O
number	O
)	O
is	O
initial	O
,	O
and	O
no	O
other	O
ordinal	B-Language
associates	O
with	O
its	O
cardinal	O
.	O
</s>
<s>
But	O
most	O
infinite	O
ordinals	B-Language
are	O
not	O
initial	O
,	O
as	O
many	O
infinite	O
ordinals	B-Language
associate	O
with	O
the	O
same	O
cardinal	O
.	O
</s>
<s>
that	O
every	O
cardinal	O
has	O
an	O
initial	O
ordinal	B-Language
.	O
</s>
<s>
In	O
theories	O
with	O
the	O
axiom	O
of	O
choice	O
,	O
the	O
cardinal	O
number	O
of	O
any	O
set	O
has	O
an	O
initial	O
ordinal	B-Language
,	O
and	O
one	O
may	O
employ	O
the	O
Von	O
Neumann	O
cardinal	O
assignment	O
as	O
the	O
cardinal	O
's	O
representation	O
.	O
</s>
<s>
(	O
However	O
,	O
we	O
must	O
then	O
be	O
careful	O
to	O
distinguish	O
between	O
cardinal	O
arithmetic	O
and	O
ordinal	B-Language
arithmetic	O
.	O
)	O
</s>
<s>
In	O
set	O
theories	O
without	O
the	O
axiom	O
of	O
choice	O
,	O
a	O
cardinal	O
may	O
be	O
represented	O
by	O
the	O
set	O
of	O
sets	O
with	O
that	O
cardinality	B-Application
having	O
minimal	O
rank	O
(	O
see	O
Scott	O
's	O
trick	O
)	O
.	O
</s>
<s>
One	O
issue	O
with	O
Scott	O
's	O
trick	O
is	O
that	O
it	O
identifies	O
the	O
cardinal	O
number	O
with	O
,	O
which	O
in	O
some	O
formulations	O
is	O
the	O
ordinal	B-Language
number	I-Language
.	O
</s>
<s>
Note	O
that	O
cardinal	O
and	O
ordinal	B-Language
arithmetic	O
agree	O
for	O
finite	O
numbers	O
.	O
</s>
<s>
The	O
α-th	O
infinite	O
initial	O
ordinal	B-Language
is	O
written	O
,	O
it	O
is	O
always	O
a	O
limit	O
ordinal	B-Language
.	O
</s>
<s>
Its	O
cardinality	B-Application
is	O
written	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
cardinality	B-Application
of	O
ω0	O
=	O
ω	B-Language
is	O
,	O
which	O
is	O
also	O
the	O
cardinality	B-Application
of	O
ω2	O
or	O
ε0	O
(	O
all	O
are	O
countable	B-Language
ordinals	I-Language
)	O
.	O
</s>
<s>
So	O
ω	B-Language
can	O
be	O
identified	O
with	O
,	O
except	O
that	O
the	O
notation	O
is	O
used	O
when	O
writing	O
cardinals	O
,	O
and	O
ω	B-Language
when	O
writing	O
ordinals	B-Language
(	O
this	O
is	O
important	O
since	O
,	O
for	O
example	O
,	O
=	O
whereas	O
)	O
.	O
</s>
<s>
Also	O
,	O
is	O
the	O
smallest	O
uncountable	O
ordinal	B-Language
(	O
to	O
see	O
that	O
it	O
exists	O
,	O
consider	O
the	O
set	O
of	O
equivalence	O
classes	O
of	O
well-orderings	O
of	O
the	O
natural	O
numbers	O
:	O
each	O
such	O
well-ordering	O
defines	O
a	O
countable	B-Language
ordinal	I-Language
,	O
and	O
is	O
the	O
order	O
type	O
of	O
that	O
set	O
)	O
,	O
is	O
the	O
smallest	O
ordinal	B-Language
whose	O
cardinality	B-Application
is	O
greater	O
than	O
,	O
and	O
so	O
on	O
,	O
and	O
is	O
the	O
limit	O
of	O
the	O
for	O
natural	O
numbers	O
n	O
(	O
any	O
limit	O
of	O
cardinals	O
is	O
a	O
cardinal	O
,	O
so	O
this	O
limit	O
is	O
indeed	O
the	O
first	O
cardinal	O
after	O
all	O
the	O
)	O
.	O
</s>
<s>
The	O
cofinality	O
of	O
an	O
ordinal	B-Language
is	O
the	O
smallest	O
ordinal	B-Language
that	O
is	O
the	O
order	O
type	O
of	O
a	O
cofinal	O
subset	O
of	O
.	O
</s>
<s>
Notice	O
that	O
a	O
number	O
of	O
authors	O
define	O
cofinality	O
or	O
use	O
it	O
only	O
for	O
limit	O
ordinals	B-Language
.	O
</s>
<s>
The	O
cofinality	O
of	O
a	O
set	O
of	O
ordinals	B-Language
or	O
any	O
other	O
well-ordered	O
set	O
is	O
the	O
cofinality	O
of	O
the	O
order	O
type	O
of	O
that	O
set	O
.	O
</s>
<s>
Thus	O
for	O
a	O
limit	O
ordinal	B-Language
,	O
there	O
exists	O
a	O
-indexed	O
strictly	O
increasing	O
sequence	O
with	O
limit	O
.	O
</s>
<s>
For	O
example	O
,	O
the	O
cofinality	O
of	O
ω2	O
is	O
ω	B-Language
,	O
because	O
the	O
sequence	O
ω·m	O
(	O
where	O
m	O
ranges	O
over	O
the	O
natural	O
numbers	O
)	O
tends	O
to	O
ω2	O
;	O
but	O
,	O
more	O
generally	O
,	O
any	O
countable	O
limit	O
ordinal	B-Language
has	O
cofinality	O
ω	B-Language
.	O
</s>
<s>
An	O
uncountable	O
limit	O
ordinal	B-Language
may	O
have	O
either	O
cofinality	O
ω	B-Language
as	O
does	O
or	O
an	O
uncountable	O
cofinality	O
.	O
</s>
<s>
And	O
the	O
cofinality	O
of	O
any	O
successor	O
ordinal	B-Language
is	O
1	O
.	O
</s>
<s>
The	O
cofinality	O
of	O
any	O
limit	O
ordinal	B-Language
is	O
at	O
least	O
.	O
</s>
<s>
An	O
ordinal	B-Language
that	O
is	O
equal	O
to	O
its	O
cofinality	O
is	O
called	O
regular	O
and	O
it	O
is	O
always	O
an	O
initial	O
ordinal	B-Language
.	O
</s>
<s>
Any	O
limit	O
of	O
regular	O
ordinals	B-Language
is	O
a	O
limit	O
of	O
initial	O
ordinals	B-Language
and	O
thus	O
is	O
also	O
initial	O
even	O
if	O
it	O
is	O
not	O
regular	O
,	O
which	O
it	O
usually	O
is	O
not	O
.	O
</s>
<s>
In	O
this	O
case	O
,	O
the	O
ordinals	B-Language
0	O
,	O
1	O
,	O
,	O
,	O
and	O
are	O
regular	O
,	O
whereas	O
2	O
,	O
3	O
,	O
,	O
and	O
ωω·2	O
are	O
initial	O
ordinals	B-Language
that	O
are	O
not	O
regular	O
.	O
</s>
<s>
The	O
cofinality	O
of	O
any	O
ordinal	B-Language
α	O
is	O
a	O
regular	O
ordinal	B-Language
,	O
i.e.	O
</s>
<s>
As	O
mentioned	O
above	O
(	O
see	O
Cantor	O
normal	O
form	O
)	O
,	O
the	O
ordinal	B-Language
ε0	O
is	O
the	O
smallest	O
satisfying	O
the	O
equation	O
,	O
so	O
it	O
is	O
the	O
limit	O
of	O
the	O
sequence	O
0	O
,	O
1	O
,	O
,	O
,	O
,	O
etc	O
.	O
</s>
<s>
Many	O
ordinals	B-Language
can	O
be	O
defined	O
in	O
such	O
a	O
manner	O
as	O
fixed	O
points	O
of	O
certain	O
ordinal	B-Language
functions	O
(	O
the	O
-th	O
ordinal	B-Language
such	O
that	O
is	O
called	O
,	O
then	O
one	O
could	O
go	O
on	O
trying	O
to	O
find	O
the	O
-th	O
ordinal	B-Language
such	O
that	O
,	O
"	O
and	O
so	O
on	O
"	O
,	O
but	O
all	O
the	O
subtlety	O
lies	O
in	O
the	O
"	O
and	O
so	O
on	O
"	O
)	O
.	O
</s>
<s>
One	O
could	O
try	O
to	O
do	O
this	O
systematically	O
,	O
but	O
no	O
matter	O
what	O
system	O
is	O
used	O
to	O
define	O
and	O
construct	O
ordinals	B-Language
,	O
there	O
is	O
always	O
an	O
ordinal	B-Language
that	O
lies	O
just	O
above	O
all	O
the	O
ordinals	B-Language
constructed	O
by	O
the	O
system	O
.	O
</s>
<s>
Perhaps	O
the	O
most	O
important	O
ordinal	B-Language
that	O
limits	O
a	O
system	O
of	O
construction	O
in	O
this	O
manner	O
is	O
the	O
Church	O
–	O
Kleene	O
ordinal	B-Language
,	O
(	O
despite	O
the	O
in	O
the	O
name	O
,	O
this	O
ordinal	B-Language
is	O
countable	O
)	O
,	O
which	O
is	O
the	O
smallest	O
ordinal	B-Language
that	O
cannot	O
in	O
any	O
way	O
be	O
represented	O
by	O
a	O
computable	O
function	O
(	O
this	O
can	O
be	O
made	O
rigorous	O
,	O
of	O
course	O
)	O
.	O
</s>
<s>
Considerably	O
large	O
ordinals	B-Language
can	O
be	O
defined	O
below	O
,	O
however	O
,	O
which	O
measure	O
the	O
"	O
proof-theoretic	O
strength	O
"	O
of	O
certain	O
formal	O
systems	O
(	O
for	O
example	O
,	O
measures	O
the	O
strength	O
of	O
Peano	O
arithmetic	O
)	O
.	O
</s>
<s>
Large	O
countable	B-Language
ordinals	I-Language
such	O
as	O
countable	O
admissible	O
ordinals	B-Language
can	O
also	O
be	O
defined	O
above	O
the	O
Church-Kleene	O
ordinal	B-Language
,	O
which	O
are	O
of	O
interest	O
in	O
various	O
parts	O
of	O
logic	O
.	O
</s>
<s>
Any	O
ordinal	B-Language
number	I-Language
can	O
be	O
made	O
into	O
a	O
topological	O
space	O
by	O
endowing	O
it	O
with	O
the	O
order	B-Algorithm
topology	I-Algorithm
;	O
this	O
topology	O
is	O
discrete	O
if	O
and	O
only	O
if	O
the	O
ordinal	B-Language
is	O
a	O
countable	O
cardinal	O
,	O
i.e.	O
</s>
<s>
at	O
most	O
ω	B-Language
.	O
</s>
<s>
A	O
subset	O
of	O
ω+1	O
is	O
open	O
in	O
the	O
order	B-Algorithm
topology	I-Algorithm
if	O
and	O
only	O
if	O
either	O
it	O
is	O
cofinite	O
or	O
it	O
does	O
not	O
contain	O
ω	B-Language
as	O
an	O
element	O
.	O
</s>
<s>
See	O
the	O
Topology	O
and	O
ordinals	B-Language
section	O
of	O
the	O
"	O
Order	B-Algorithm
topology	I-Algorithm
"	O
article	O
.	O
</s>
<s>
The	O
transfinite	B-Language
ordinal	I-Language
numbers	I-Language
,	O
which	O
first	O
appeared	O
in	O
1883	O
,	O
originated	O
in	O
Cantor	O
's	O
work	O
with	O
derived	O
sets	O
.	O
</s>
<s>
To	O
define	O
this	O
set	O
,	O
he	O
defined	O
the	O
transfinite	B-Language
ordinal	I-Language
numbers	I-Language
and	O
transformed	O
the	O
infinite	O
indices	O
into	O
ordinals	B-Language
by	O
replacing	O
∞	O
with	O
ω	B-Language
,	O
the	O
first	O
transfinite	B-Language
ordinal	I-Language
number	I-Language
.	O
</s>
<s>
Cantor	O
called	O
the	O
set	O
of	O
finite	O
ordinals	B-Language
the	O
first	O
number	O
class	O
.	O
</s>
<s>
The	O
second	O
number	O
class	O
is	O
the	O
set	O
of	O
ordinals	B-Language
whose	O
predecessors	O
form	O
a	O
countably	O
infinite	O
set	O
.	O
</s>
<s>
The	O
set	O
of	O
all	O
α	O
having	O
countably	O
many	O
predecessors	O
—	O
that	O
is	O
,	O
the	O
set	O
of	O
countable	B-Language
ordinals	I-Language
—	O
is	O
the	O
union	O
of	O
these	O
two	O
number	O
classes	O
.	O
</s>
<s>
Cantor	O
proved	O
that	O
the	O
cardinality	B-Application
of	O
the	O
second	O
number	O
class	O
is	O
the	O
first	O
uncountable	O
cardinality	B-Application
.	O
</s>
<s>
Cantor	O
's	O
second	O
theorem	O
becomes	O
:	O
If	O
P′	O
is	O
countable	O
,	O
then	O
there	O
is	O
a	O
countable	B-Language
ordinal	I-Language
α	O
such	O
that	O
.	O
</s>
<s>
Therefore	O
,	O
there	O
is	O
a	O
countable	B-Language
ordinal	I-Language
α	O
such	O
that	O
.	O
</s>
<s>
Cantor	O
's	O
work	O
with	O
derived	O
sets	O
and	O
ordinal	B-Language
numbers	I-Language
led	O
to	O
the	O
Cantor-Bendixson	O
theorem	O
.	O
</s>
<s>
Using	O
successors	O
,	O
limits	O
,	O
and	O
cardinality	B-Application
,	O
Cantor	O
generated	O
an	O
unbounded	O
sequence	O
of	O
ordinal	B-Language
numbers	I-Language
and	O
number	O
classes	O
.	O
</s>
<s>
The	O
-th	O
number	O
class	O
is	O
the	O
set	O
of	O
ordinals	B-Language
whose	O
predecessors	O
form	O
a	O
set	O
of	O
the	O
same	O
cardinality	B-Application
as	O
the	O
α-th	O
number	O
class	O
.	O
</s>
<s>
The	O
cardinality	B-Application
of	O
the	O
-th	O
number	O
class	O
is	O
the	O
cardinality	B-Application
immediately	O
following	O
that	O
of	O
the	O
α-th	O
number	O
class	O
.	O
</s>
<s>
For	O
a	O
limit	O
ordinal	B-Language
α	O
,	O
the	O
α-th	O
number	O
class	O
is	O
the	O
union	O
of	O
the	O
β-th	O
number	O
classes	O
for	O
.	O
</s>
<s>
Its	O
cardinality	B-Application
is	O
the	O
limit	O
of	O
the	O
cardinalities	B-Application
of	O
these	O
number	O
classes	O
.	O
</s>
<s>
If	O
n	O
is	O
finite	O
,	O
the	O
n-th	O
number	O
class	O
has	O
cardinality	B-Application
.	O
</s>
<s>
If	O
,	O
the	O
α-th	O
number	O
class	O
has	O
cardinality	B-Application
.	O
</s>
<s>
Therefore	O
,	O
the	O
cardinalities	B-Application
of	O
the	O
number	O
classes	O
correspond	O
one-to-one	O
with	O
the	O
aleph	O
numbers	O
.	O
</s>
<s>
Also	O
,	O
the	O
α-th	O
number	O
class	O
consists	O
of	O
ordinals	B-Language
different	O
from	O
those	O
in	O
the	O
preceding	O
number	O
classes	O
if	O
and	O
only	O
if	O
α	O
is	O
a	O
non-limit	O
ordinal	B-Language
.	O
</s>
<s>
Therefore	O
,	O
the	O
non-limit	O
number	O
classes	O
partition	O
the	O
ordinals	B-Language
into	O
pairwise	O
disjoint	O
sets	O
.	O
</s>
